of the Principle of Least Action. 431 



plane, 2c -r{pr}.hr is the force with which these parts are 



pressed together ; and this force is also equal to 2qc8r } where c is 

 the length of the cylinder, so that 



q = Tr {pr\=P + r dr . 



Now the work done is by Clapeyron's theorem proportional 



to \{(p + qf—kpq}rdr (since p and q are principal pressures) ^ 



k being a constant depending on the nature of the material*. 

 We must therefore have 



\f{2{p + q)-kq}8p+{2{p + q)--kp}8q]rdr=0 



for all forms of 8p 8q consistent with 



Sq = Sp + r.-^ } 



whence, by the rules of the calculus of variations, 



d 

 dr 



{2(p+q)-kq-\}r + l{\r*}=0, 



2{p + q)-kp + \=0, 

 where \ is an unknown function of r. Eliminating \, we obtain 



{4,-k\ {p + q)=j r {2r\p + q)-kpr*\; 



but 



p + q=2p + rf = l£ r {pr*\; 



■■^)i{^}=i{^ d £+^-k)pr*}-, 



whence 

 and 



2r 3 4- = constant, 

 dr 



v= -I +c q 



Q=-%+ c *> 



* The work done in a parallelopipedal element will not be altered by a 

 quantity of the same order by supposing its sides inclined at an infinitesimal 

 angle. The third principal pressure is parallel to the axis of the cylinder, 

 and may be supposed zero without at all affecting the result. 



